In Exercise 14.5(b), suppose you predict \(Y\) using \(\bar{Y} / 2\) instead of \(\bar{Y}\).

a. Compute the bias of the prediction.

b. Compute the mean of the prediction error.

c. Compute the variance of the prediction error.

d. Compute the MSPE of the prediction.

e. Does \(\bar{Y} / 2\) produce a prediction with a lower MSPE than the \(\bar{Y}\) prediction?

f. Suppose \(\mu=10\) (instead of \(\mu=2\) ). Does \(\bar{Y} / 2\) produce a prediction with a lower MSPE than the \(\bar{Y}\) prediction?

g. In a realistic setting, the value of \(\mu\) is unknown. What advice would you give someone who is deciding between using \(\bar{Y}\) and \(\bar{Y} / 2\)?

Exercise 14.5(b)

\(Y\) is a random variable with mean \(\mu=2\) and variance \(\sigma^{2}=25\).

b. Suppose you don't know the value of \(\mu\) but you have access to a random sample of size \(n=10\) from the same population. Let \(\bar{Y}\) denote the sample mean from this random sample. You predict the value of \(Y\) using \(\bar{Y}\).

i. Show that the prediction error can be decomposed as \(Y-\bar{Y}=(Y-\mu)-\) \((\bar{Y}-\mu)\), where \((Y-\mu)\) is the prediction error of the oracle predictor and \((\mu-\bar{Y})\) is the error associated with using \(\bar{Y}\) as an estimate of \(\mu\).

ii. Show that \((Y-\mu)\) has a mean of 0 , that \((\bar{Y}-\mu)\) has a mean of 0 , and that \(Y-\bar{Y}\) has a mean of 0 .

iii. Show that \((Y-\mu)\) and \((\bar{Y}-\mu)\) are uncorrelated.

iv. Show that the MSPE of \(\bar{Y}\) is MSPE \(=E(Y-\mu)^{2}+E(\bar{Y}-\mu)^{2}=\) \(\operatorname{var}(Y)+\operatorname{var}(\bar{Y})\).

v. Show that MSPE \(=25(1+1 / 10)=27.5\).